<p>Let <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mathscr {A}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">A</mi> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mathscr {B}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">B</mi> </math></EquationSource> </InlineEquation> be two algebras, and let <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\sigma , \tau :\mathscr {A} \rightarrow \mathscr {B}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>σ</mi> <mo>,</mo> <mi>τ</mi> <mo>:</mo> <mi mathvariant="script">A</mi> <mo stretchy="false">→</mo> <mi mathvariant="script">B</mi> </mrow> </math></EquationSource> </InlineEquation> be two linear mappings. A linear mapping <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$d_1:\mathscr {A} \rightarrow \mathscr {B}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>d</mi> <mn>1</mn> </msub> <mo>:</mo> <mi mathvariant="script">A</mi> <mo stretchy="false">→</mo> <mi mathvariant="script">B</mi> </mrow> </math></EquationSource> </InlineEquation> is called a <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$(\sigma , \tau )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>σ</mi> <mo>,</mo> <mi>τ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-ternary derivation if there exist the linear mappings <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$d_2, d_3:\mathscr {A} \rightarrow \mathscr {B}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>d</mi> <mn>2</mn> </msub> <mo>,</mo> <msub> <mi>d</mi> <mn>3</mn> </msub> <mo>:</mo> <mi mathvariant="script">A</mi> <mo stretchy="false">→</mo> <mi mathvariant="script">B</mi> </mrow> </math></EquationSource> </InlineEquation> which satisfy <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$d_1(ab) = d_2(a) \sigma (b) + \tau (a)d_3(b)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>d</mi> <mn>1</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mi>b</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msub> <mi>d</mi> <mn>2</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> <mi>σ</mi> <mrow> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mrow> <mo>+</mo> <mi>τ</mi> <mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> <msub> <mi>d</mi> <mn>3</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> for all <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$a, b \in \mathscr {A}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>∈</mo> <mi mathvariant="script">A</mi> </mrow> </math></EquationSource> </InlineEquation>. By a <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$(d_2, \sigma , \tau , d_3)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <msub> <mi>d</mi> <mn>2</mn> </msub> <mo>,</mo> <mi>σ</mi> <mo>,</mo> <mi>τ</mi> <mo>,</mo> <msub> <mi>d</mi> <mn>3</mn> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-derivation, we mean a <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$(\sigma , \tau )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>σ</mi> <mo>,</mo> <mi>τ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-ternary derivation <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$d_1$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>d</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation> associated with the mappings <InlineEquation ID="IEq14"> <EquationSource Format="TEX">$d_2$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>d</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq15"> <EquationSource Format="TEX">$d_3$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>d</mi> <mn>3</mn> </msub> </math></EquationSource> </InlineEquation>. The main purpose of this article is to study the characterization and automatic continuity of such derivations on Banach algebras and <InlineEquation ID="IEq16"> <EquationSource Format="TEX">$C^{*}$</EquationSource> <EquationSource Format="MATHML"><math> <mmultiscripts> <mi>C</mi> <mrow /> <mrow> <mrow /> <mo>∗</mo> </mrow> </mmultiscripts> </math></EquationSource> </InlineEquation>-algebras.</p>

On the characterization and automatic continuity of $(\sigma , \tau )$-ternary derivations

期刊论文
发表日期：

2025年12月12日

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Amin Hosseini,
Choonkil Park,
Mehdi Mohammadzadeh Karizaki

摘要

Let $\mathscr {A}$ $A$ and $\mathscr {B}$ $B$ be two algebras, and let $\sigma , \tau :\mathscr {A} \rightarrow \mathscr {B}$ $σ, τ : A \to B$ be two linear mappings. A linear mapping $d_1:\mathscr {A} \rightarrow \mathscr {B}$ $d_{1} : A \to B$ is called a $(\sigma , \tau )$ $(σ, τ)$ -ternary derivation if there exist the linear mappings $d_2, d_3:\mathscr {A} \rightarrow \mathscr {B}$ $d_{2}, d_{3} : A \to B$ which satisfy $d_1(ab) = d_2(a) \sigma (b) + \tau (a)d_3(b)$ $d_{1} (a b) = d_{2} (a) σ (b) + τ (a) d_{3} (b)$ for all $a, b \in \mathscr {A}$ $a, b \in A$ . By a $(d_2, \sigma , \tau , d_3)$ $(d_{2}, σ, τ, d_{3})$ -derivation, we mean a $(\sigma , \tau )$ $(σ, τ)$ -ternary derivation $d_1$ $d_{1}$ associated with the mappings $d_2$ $d_{2}$ and $d_3$ $d_{3}$ . The main purpose of this article is to study the characterization and automatic continuity of such derivations on Banach algebras and $C^{*}$ $C^{*}$ -algebras.

On the characterization and automatic continuity of \((\sigma , \tau )\)-ternary derivations

摘要